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arxiv: 2607.02286 · v1 · pith:EZJIHWJ7new · submitted 2026-07-02 · 🧮 math.NA · cs.NA

A novel time-domain iterative method for a three-dimensional inverse acoustic obstacle scattering problem

Lu Zhao , Heping Dong , Zhiyong Cheng This is my paper

Pith reviewed 2026-07-03 07:44 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords inverse acoustic scatteringtime domainconvolution quadraturehomothetic surfaceobstacle reconstructionretarded potentialiterative methodFréchet derivative
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The pith

A retarded boundary integral on a homothetic surface enables iterative reconstruction of 3D rigid acoustic obstacles in the time domain.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a time-domain iterative method for recovering the shape and location of a rigid obstacle from acoustic scattering measurements. It defines retarded boundary integrals on a homothetic surface and discretizes the forward problem with convolution quadrature and Galerkin methods. These integrals are recast in the s-domain to yield only non-singular computations, while the Fréchet derivative with respect to the boundary is obtained directly for use in the iteration. The authors prove that the field produced by the homothetic surface converges to the true scattered field and introduce incremental truncation to stabilize the inversion. Numerical tests are presented to illustrate the method's performance.

Core claim

By introducing the retarded boundary integral defined on a homothetic surface, a novel time-domain convolution quadrature based iterative method is proposed to reconstruct both the shape and location of a rigid obstacle. The retarded integral in the time domain is reformulated into a system of integrals in the s-domain. The resulting s-domain integrals are very fast to compute, as they only involve non-singular integrals over the homothetic surfaces. Moreover, the Fréchet derivative with respect to the boundary can be derived straightforwardly. We also prove that the scattered field generated by the homothetic surface converges to the exact field in the time domain. To improve the stability

What carries the argument

The retarded boundary integral defined on a homothetic surface, which carries the reconstruction by allowing fast non-singular s-domain evaluation and direct access to the Fréchet derivative.

If this is right

  • The algorithm recovers both the shape and the location of the rigid obstacle.
  • All integrals reduce to non-singular evaluations over homothetic surfaces and are therefore fast to compute.
  • The Fréchet derivative with respect to the boundary is obtained in a straightforward manner.
  • Incremental truncation stabilizes the iterative inversion.
  • Numerical experiments demonstrate effectiveness and robustness for the tested cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The s-domain reformulation may lower computational cost relative to direct time-domain boundary integrals in other inverse wave problems.
  • The homothetic-surface construction could be tested on related time-domain inverse problems that currently require singular-integral handling on the unknown boundary.
  • The convergence result supplies a concrete justification for replacing the true boundary with a nearby homothetic proxy during early iterations.

Load-bearing premise

The scattered field generated by the homothetic surface converges to the exact field in the time domain.

What would settle it

A numerical simulation or analytic counterexample in which the field produced by the homothetic surface fails to approach the exact scattered field as the homothety parameter tends to the true boundary.

Figures

Figures reproduced from arXiv: 2607.02286 by Heping Dong, Lu Zhao, Zhiyong Cheng.

Figure 1
Figure 1. Figure 1: An illustration of acoustic obstacle scattering. Time-domain broadband signals typically contain richer information and are often easier to capture in practice than frequency-domain data. In recent years, significant mathematical and computational progress has been achieved on time-domain acoustic obstacle scattering and the associated inverse scattering problems. Concerning the well-posedness analysis of … view at source ↗
Figure 2
Figure 2. Figure 2: Verification of the forward scheme using the cushion-shaped obstacle [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstructions of a pinched ball-shaped obstacle with different levels of noise. The initial guess is a sphere with c (0) = (−0.5, 0.4, −0.3)⊤ and r (0) = 0.6. The incident point source is located at (0, 0, 5)⊤. regularization scheme based on homothetic surfaces. To test stability, the noisy data u sc δ˜ (x, t) is generated as follows u sc δ˜ = (1 + ˜δΘ)u sc D with Θ being normally distributed random vari… view at source ↗
Figure 4
Figure 4. Figure 4: Reconstructions of a cushion-shaped obstacle with different levels of noise. The initial guess is a sphere with c (0) = (−0.3, 0.2, −0.3)⊤ and r (0) = 0.5. The incident point source is located at (0, 0, 5)⊤. with Hγ (γ = 1/2) penalty term, and the regularization parameters α in (19) and λ are set to a constant value of 10−8 and 10−2 . In addition, we adopt the incident wave u inc(x, t) =    1000 sin(4… view at source ↗
Figure 5
Figure 5. Figure 5: Reconstructions of a pinched ball-shaped obstacle with multiple launch positions. The initial guess is a sphere with c (0) = (−0.5, 0.4, −0.3)⊤ and r (0) = 0.6. (a) Reconstruction with 10% noise. The incident point sources are located at (0, 0, 5)⊤ and (0, 0, −5)⊤. (b) Reconstruction with 10% noise. The incident point sources are located at (0, 5, 0)⊤, (0, −5, 0)⊤, (0, 0, 5)⊤ and (0, 0, −5)⊤ [PITH_FULL_IM… view at source ↗
Figure 6
Figure 6. Figure 6: Reconstructions of a cushion-shaped obstacle with multiple launch posi￾tions. The initial guess is a sphere with c (0) = (−0.3, 0.2, −0.3)⊤ and r (0) = 0.5. The initial guess is taken as a sphere with center c (0) and radius r (0). We choose the scaling factor ρ = 0.5, the geometric contraction factor ς = 0.9, and set the number of inner cycles per l to be loop = 2. The number of quadrature points on ΓD′ i… view at source ↗
Figure 7
Figure 7. Figure 7: Reconstructions of a pinched ball-shaped obstacle with 10% noise, where the incident point sources are located at (0, 0, 5)⊤ and (0, 0, −5)⊤. become extremely small and contribute little to improving the reconstruction [6]. Therefore, in our implementation, we skip those wavenumbers for which the data magnitude is below a prescribed threshold, and continue the iteration with the remaining complex wavenumbe… view at source ↗
Figure 8
Figure 8. Figure 8: Reconstructions of a cushion-shaped obstacle with 10% noise, where the incident point sources are located at (0, 0, 5)⊤ and (0, 0, −5)⊤. To improve the reconstruction accuracy, we investigate multiple launch positions of the incident wave. In this example, the reconstructions are carried out using scattered field data contaminated with 10% noise for two and four incident point sources. For comparison, the … view at source ↗
Figure 9
Figure 9. Figure 9: Reconstructions of a bean-shaped obstacle with 1% noise. The initial guess is a sphere with c (0) = (−0.2, 0.3, −0.4)⊤ and r (0) = 0.7 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Reconstruction of a complex obstacle from time-domain scattered field data generated by four point sources located at (0, 0, 5)⊤, (0, 0, −5)⊤, (5, 0, 0)⊤, and (−5, 0, 0)⊤, under different noise levels. c (0) = (−0.3, 0.2, −0.5)⊤, r (0) = 0.3. (a) Reconstruction with 1% noise [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Reconstruction of a complex obstacle obtained by directly increasing the truncation number from M = 0 to M = 8 without incremental truncation technique. The time-domain scattered field data are generated by four point sources located at (0, 0, 5)⊤, (0, 0, −5)⊤, (5, 0, 0)⊤, and (−5, 0, 0)⊤. c (0) = (−0.3, 0.2, −0.5)⊤, r (0) = 0.3 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Reconstruction of a complex obstacle from time-domain scattered field data generated by four point sources located at (0, 0, 5)⊤, (0, 0, −5)⊤,(5, 0, 0)⊤ and (−5, 0, 0)⊤ with 1% noise. c (0) = (−0.3, 0.2, −0.5)⊤, r (0) = 0.3. Subfigures (a)–(f) show the initial surface and the progressive reconstructions obtained as the truncation number M increases. the overall shape is reasonably captured, some fine-scal… view at source ↗
read the original abstract

This paper concerns the three-dimensional forward and inverse acoustic obstacle scattering problem in the time domain. For the forward problem, a retarded potential formulation discretized by convolution quadrature and Galerkin methods is introduced. By introducing the retarded boundary integral defined on a homothetic surface, we propose a novel time-domain convolution quadrature based iterative method to reconstruct both the shape and location of a rigid obstacle. The retarded integral in the time domain is reformulated into a system of integrals in the s-domain. The resulting s-domain integrals are very fast to compute, as they only involve non-singular integrals over the homothetic surfaces. Moreover, the Fr\'echet derivative with respect to the boundary can be derived straightforwardly. We also prove that the scattered field generated by the homothetic surface converges to the exact field in the time domain. To improve the stability of the inversion algorithm, an incremental truncation technique is proposed, and numerical experiments confirm the effectiveness and robustness of our method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript addresses the three-dimensional forward and inverse acoustic obstacle scattering problem in the time domain. For the forward problem it introduces a retarded potential formulation discretized by convolution quadrature and Galerkin methods. For the inverse problem it proposes a novel iterative method based on retarded boundary integrals defined on a homothetic surface; the integrals are reformulated in the s-domain to yield non-singular computations, the Fréchet derivative with respect to the boundary is derived, convergence of the homothetic-surface scattered field to the exact field is proved, an incremental truncation technique is introduced for stability, and numerical experiments are presented to illustrate effectiveness and robustness.

Significance. If the convergence proof and the supporting analysis hold, the work offers a concrete advance in time-domain inverse scattering by replacing singular integrals with non-singular ones on homothetic surfaces and by supplying an explicit Fréchet derivative together with a stability device. The combination of a proved convergence result, an s-domain reformulation that accelerates evaluation, and numerical confirmation of robustness constitutes a substantive contribution to the numerical analysis of inverse obstacle problems.

minor comments (3)
  1. The abstract states that the s-domain integrals 'only involve non-singular integrals over the homothetic surfaces,' but the precise statement of the singularity removal (e.g., the distance between the two homothetic surfaces) should be made explicit in the first paragraph of §3 or §4 so that readers can verify the claim without searching later sections.
  2. Notation for the homothetic scaling parameter (denoted variously as λ or α in the abstract and later text) should be unified and introduced once in §2 before its repeated use in the convergence argument.
  3. The incremental truncation technique is mentioned in the abstract and claimed to improve stability, yet no reference is given to the specific truncation threshold or the section where its effect on the iteration is quantified; a short paragraph or table entry would clarify its implementation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the assessment of its significance, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation chain rests on an explicit proof that the scattered field generated by the homothetic surface converges to the exact field in the time domain; this convergence result is stated as independently established rather than assumed or fitted. The retarded-potential forward formulation, s-domain reformulation, Fréchet derivative, and incremental truncation are presented as separate technical steps whose validity does not reduce to the target inverse reconstruction by construction. No self-definitional relations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the given description, so the central iterative method remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no details on specific free parameters, axioms, or invented entities beyond standard assumptions in boundary integral methods for scattering problems.

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Reference graph

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